Resolvent-based optimization for approximating the statistics of a chaotic Lorenz system.

IF 2.2 3区 物理与天体物理 Q2 PHYSICS, FLUIDS & PLASMAS
Thomas Burton, Sean Symon, Ati S Sharma, Davide Lasagna
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Abstract

We propose a framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional approaches relying on identifying all short, fundamental unstable periodic orbits to compute ergodic averages via cycle expansion are computationally prohibitive for high-dimensional fluid systems. Our framework stems from the observation in Lasagna [D. Lasagna, Phys. Rev. E 102, 052220 (2020)2470-004510.1103/PhysRevE.102.052220] that a single unstable periodic orbit with a period sufficiently long to span a large fraction of the attractor captures the statistical properties of chaotic trajectories. Given the difficulty of identifying unstable periodic orbits for high-dimensional fluid systems, approximate trajectories residing in a low-dimensional subspace are instead constructed using resolvent modes, which inherently capture the temporal periodicity of unstable periodic orbits. The amplitude coefficients of these modes are adjusted iteratively with gradient-based optimization to minimize the violation of the projected governing equations, producing trajectories that approximate, rather than exactly solve, the system dynamics. An attempt at utilizing this framework on a chaotic system is made here on the Lorenz 1963 equations, where resolvent analysis enables an exact dimensionality reduction from three to two dimensions. Key observables averaged over these trajectories produced by the approach as well as probability distributions and spectra rapidly converge to values obtained from long chaotic simulations, even with a limited number of iterations. This indicates that exact solutions may not be necessary to approximate the system's statistical behavior, as the trajectories obtained from partial optimization provide a sufficient "sketch" of the attractor in state space.

混沌洛伦兹系统统计量近似的求解优化。
我们提出了一个框架,通过结合变分方法来搜索不稳定周期轨道和解析分析来降低维数,近似湍流的统计性质。传统的方法依赖于识别所有短的、基本的不稳定周期轨道,通过循环展开来计算遍历平均,这对于高维流体系统来说在计算上是禁止的。我们的框架源于对千层面的观察[D。烤宽面条、phy。[j] .物理学报,1999,19 (4):555 - 555 . [j] .物理学报,1999,19(5):555 - 555。考虑到识别高维流体系统的不稳定周期轨道的困难,使用固有捕获不稳定周期轨道的时间周期性的解析模式来构建驻留在低维子空间中的近似轨迹。这些模式的振幅系数通过基于梯度的优化迭代调整,以最大限度地减少对投影控制方程的违反,从而产生近似而不是精确求解系统动力学的轨迹。在洛伦兹1963方程中,我们尝试在混沌系统中利用这一框架,其中可解性分析能够精确地从三维降维到二维。该方法产生的这些轨迹上的关键观测值的平均值以及概率分布和光谱迅速收敛于从长时间混沌模拟中获得的值,即使迭代次数有限。这表明,精确的解可能不需要近似系统的统计行为,因为从部分优化中获得的轨迹提供了状态空间中吸引子的充分“草图”。
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来源期刊
Physical Review E
Physical Review E PHYSICS, FLUIDS & PLASMASPHYSICS, MATHEMAT-PHYSICS, MATHEMATICAL
CiteScore
4.50
自引率
16.70%
发文量
2110
期刊介绍: Physical Review E (PRE), broad and interdisciplinary in scope, focuses on collective phenomena of many-body systems, with statistical physics and nonlinear dynamics as the central themes of the journal. Physical Review E publishes recent developments in biological and soft matter physics including granular materials, colloids, complex fluids, liquid crystals, and polymers. The journal covers fluid dynamics and plasma physics and includes sections on computational and interdisciplinary physics, for example, complex networks.
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